cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A305134 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( 2*exp(n*x) - A(x) )^n / 2^(n+1).

Original entry on oeis.org

1, 6, 106, 9798, 2042986, 721198086, 378754904746, 274462194065478, 261211828432706026, 315282684090141417606, 470124979835875652863786, 848422945353825106452994758, 1822526603267557240862350671466, 4596139606368556055825161023870726, 13448584326250762088160567798167642026, 45199506338787031550197525974862852621638
Offset: 0

Views

Author

Paul D. Hanna, May 29 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + 6*x + 106*x^2/2! + 9798*x^3/3! + 2042986*x^4/4! + 721198086*x^5/5! + 378754904746*x^6/6! + 274462194065478*x^7/7! + 261211828432706026*x^8/8! + 315282684090141417606*x^9/9! + 470124979835875652863786*x^10/10! + ...
such that
1 = 1/2  +  (2*exp(x) - A(x))/2^2  +  (2*exp(2*x) - A(x))^2/2^3  +  (2*exp(3*x) - A(x))^3/2^4  +  (2*exp(4*x) - A(x))^4/2^5  +  (2*exp(5*x) - A(x))^5/2^6 + ...
Also,
1 = 1/(2 + A(x))  +  2*exp(x)/(2 + exp(x)*A(x))^2  +  2^2*exp(4*x)/(2 + exp(2*x)*A(x))^3  +  2^3*exp(9*x)/(2 + exp(3*x)*A(x))^4  +  2^4*exp(16*x)/(2 + exp(4*x)*A(x))^5  +  2^5*exp(25*x)/(2 + exp(5*x)*A(x))^6  + ...
RELATED SERIES.
log(A(x)) = 6*x + 70*x^2/2! + 8322*x^3/3! + 1812142*x^4/4! + 657412530*x^5/5! + 351254035150*x^6/6! + 257586196964082*x^7/7! + 247297892785673422*x^8/8! + 300478711708843324530*x^9/9! + 450397140484880214948430*x^10/10! + ...

Links

Crossrefs

Cf. A304640, A301436.

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) 1 = Sum_{n>=0} ( 2*exp(n*x) - A(x) )^n / 2^(n+1).
(2) 1 = Sum_{n>=0} 2^n * exp(n^2*x) / (2 + exp(n*x) * A(x))^(n+1).

A304641 E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n.

Original entry on oeis.org

1, 2, 6, 74, 3078, 228842, 25277286, 3837501194, 762731347398, 191798593122602, 59475206565622566, 22290155840476400714, 9933314218291366691718, 5192540728710234994272362, 3147427468437058629798524646, 2190237887318737512524514442634, 1734606000858253287464231519860038, 1551466530739217915273113571521758122, 1556475858078242120174483544923467343526
Offset: 0

Views

Author

Paul D. Hanna, May 16 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 74*x^3/3! + 3078*x^4/4! + 228842*x^5/5! + 25277286*x^6/6! + 3837501194*x^7/7! + 762731347398*x^8/8! + 191798593122602*x^9/9! + 59475206565622566*x^10/10! + ...
such that
1 = 1  +  (exp(2*x) - A(x))  +  (exp(3*x) - A(x))^2  +  (exp(4*x) - A(x))^3  +  (exp(5*x) - A(x))^4  +  (exp(6*x) - A(x))^5  +  (exp(7*x) - A(x))^6  +  (exp(8*x) - A(x))^7 + ...
Also,
1 = 1/(1 + A(x))  +  exp(2*x)/(1 + exp(x)*A(x))^2  +  exp(6*x)/(1 + exp(2*x)*A(x))^3  +  exp(12*x)/(1 + exp(3*x)*A(x))^4  +  exp(20*x)/(1 + exp(4*x)*A(x))^5  +  exp(30*x)/(1 + exp(5*x)*A(x))^6  +  exp(42*x)/(1 + exp(6*x)*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = 2*x + 2*x^2/2! + 54*x^3/3! + 2570*x^4/4! + 199590*x^5/5! + 22598762*x^6/6! + 3488755494*x^7/7! + 701959131050*x^8/8! + 178186466260710*x^9/9! + 55669778154059882*x^10/10! + ...
exp(-x) * A(x) = 1 + x + 3*x^2/2! + 61*x^3/3! + 2811*x^4/4! + 214141*x^5/5! + 23949003*x^6/6! + 3665260621*x^7/7! + 732726498171*x^8/8! + 185070066199261*x^9/9! + 57591088296085803*x^10/10! + ...

Crossrefs

Cf. A304640, A304642.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (exp((m+1)*x +x*O(x^#A)) - Ser(A))^m ) )[#A] ); n!*A[n+1]}
for(n=0,20, print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n.
(2) 1 = Sum_{n>=0} exp(n*(n+1)*x) / (1 + exp(n*x)*A(x))^(n+1).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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